We study a $ϕ^4$-theory at finite temperature in a finite volume. Quantum, thermal and volume fluctuations are treated with the functional renormalisation group. Specifically, we focus on the interplay of temperature and length scales driving the system. We find that thermodynamical observables at finite volume such as the pressure approach the infinite volume limit similarly to that of the vanishing temperature limit. We also advance the functional renormalisation group method at finite volume. In particular, we identify requirements for suitable regulators that admit the exponential thermal and finite volume decay properties.
We apply the Local Discontinuous Galerkin discretisation to flow equations of the O(N)-model in the Local Potential Approximation. The improved stability is directly observed by solving the flow equation for various $N$ and space-time dimensions $d$. A particular focus of this work is the numerical discretisation and its implementation. The code is publicly available, and is explained in detail here. It is realised as a module within the high performance PDE framework DUNE.
We put forward a functional renormalisation group approach for the direct computation of real time correlation functions, also applicable at finite temperature and density. We construct a general class of regulators that preserve the space-time symmetries, and allows the computation of correlation functions at complex frequencies. This includes both imaginary time and real time, and allows in particular the use of the plethora of imaginary time results for the computation of real time correlation functions. We also discuss real time computation on the Keldysh contour with general spatial momen…
We discuss an approach for accessing bound state properties, like mass and decay width, of a theory within the functional renormalisation group approach. An important cornerstone is the dynamical hadronization technique for resonant interaction channels. The general framework is exemplified and put to work within the two-flavour quark-meson model. This model provides a low-energy description of the dynamics of two-flavour QCD with quark and hadronic degrees of freedom. We compare explicitly the respective results for correlation functions and observables with first principle QCD results in a q…
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心与物理学院）；Institut für Theoretische Physik, Universität Heidelberg（海德堡大学理论物理研究所）；ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung mbH（GSI重离子研究中心极端物质研究所EMMI）；Department of Physics and Astronomy, Kwansei Gakuin University（关西学院大学物理天文学系）
The Local Potential Approximation (LPA) to the Wetterich-equation is formulated explicitly in terms of operators, which are invariant under the $U(n)\times U(n)$ symmetry group. Complete formulas are presented for the two-flavor ($U(2)\times U(2)$) case. The same approach leads to a unique natural truncation of the functional driving the renormalisation flow of the potential of the three-flavor case ($U(3)\times U(3)$). The procedure applied to the $SU(3)\times SU(3)$ symmetric theory, results in an equation, which potentially allows an RG-investigation of the effect of the 't Hooft term repre…
The scalar mass is determined in the simplest scalar-fermion Yukawa-model in the whole range of stability of the scalar potential. Two versions of the Functional Renormalisation Group (FRG) equations are solved, where also composite fermionic background is taken into account. The close agreement of the results with previous studies taking into account exclusively the effect of the scalar condensate, supports a rather small systematic truncation error of FRG due to the omission of higher dimensional operators.
Functional Renormalisation Group (FRG) equations are constructed for a simple Yukawa-model with discrete chiral symmetry, including also the effect of a nonzero composite fermion background beyond the conventional scalar condensate. The evolution of the effective potential of the model, generically depending on two invariants, is explored with help of power series expansions. Systematic investigation of the effect of a class of irrelevant operators on the lower (stability) bound allows a non-perturbative extension of the maximal cut-off value consistent with any given mass of the scalar field.
We develop a non-perturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularisation. Therefore, the non-perturbative spectral renormalisation setup here respects all symmetries of the theories at hand. In particular this includes space-time symmetries as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalisation can be applied within general functional approaches such as the functional renormalisation…
We examine the precise connection between the exact renormalisation group with local couplings and the renormalisation of correlation functions of composite operators in scale-invariant theories. A geometric description of theory space allows us to select convenient non-linear parametrisations that serve different purposes. First, we identify normal parameters in which the renormalisation group flows take their simplest form; normal correlators are defined by functional differentiation with respect to these parameters. The renormalised correlation functions are given by the continuum limit of …
We study the phase diagram of two-color Quantum Chromodynamics at finite temperature and chemical potential. This is done within an effective low-energy description in terms of quarks, mesons and diquarks. Quantum, thermal and density fluctuations are taken into account with the functional renormalisation group approach. In particular, we establish the phenomenon of pre-condensation, affecting the location of the phase boundary to Bose-Einstein condensation. We also discuss the Silver Blaze property in the context of the functional renormalisation group.
We compute non-perturbative spectral functions in a scalar $ϕ^4$-theory in three spacetime dimensions via the spectral functional renormalisation group. This approach allows for the direct, manifestly Lorentz covariant computation of correlation functions in Minkowski spacetime, including a physical on-shell renormalisation. We present numerical results for the spectral functions of the two- and four-point correlation functions for different values of the coupling parameter. These results agree very well with those obtained from another functional real-time approach, the spectral Dyson-Schwing…
Functional renormalisation group approach is applied to a system of kaons with finite chemical potential. A set of approximate flow equations for the effective couplings is derived and solved. At high scale the system is found to be at the normal phase whereas at some critical value of the running scale it undergoes the phase transition (PT) to the phase with a spontaneously broken symmetry with the kaon condensate as an order parameter. The value of the condensate turns out to be quite sensitive to the kaon-kaon scattering length.
We discuss the far-from-equilibrium evolution of $ϕ^3$-theory in $1+1$ dimensions with the temporal functional renormalisation group \cite{Gasenzer:2007za, Gasenzer:2010rq}. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behaviour suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conse…
Equivalence criteria are established for an effective Yukawa-type theory of composite fields representing two-particle fermion bound states with the original "microscopic" theory of interacting fermions based on the spectral decomposition of the 2-to-2 fermion scattering amplitude. Functional renormalisation group equations of the effective theory are derived exploiting relations expressing the equivalence. The effect of truncating the spectral decomposition is investigated quantitatively on the example of the non-relativistic bound states of two oppositely charged fermi particles.
We present a comprehensive study of the quark sector of $2+1$ flavour QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with $2+1$ dynamical quark flavours, obtained from studies with Schwinger-Dyson equations, the functional renormalisation group approach, …
The renormalisation group flow of the chiral Nambu--Jona-Lasinio (NJL) model with one fermion flavor is mapped out in the symmetric phase with the help of the Functional Renormalisation Group (FRG) method using a physically motivated non-local trial effective action. The well-known infrared unstable strongly coupled fixed point characterized by a set of pointlike four-fermion couplings is reproduced. The Gaussian infrared end-point of the flow of the four-fermion couplings is now accompanied by non-zero limiting composite couplings characteristic for interacting two-particle states with finite…
We discuss the QCD phase structure at finite temperature and chemical potential for $2$-flavour and $2+1$-flavour QCD. The results are achieved by computing QCD correlation functions within a generalised functional approach that combines Dyson-Schwinger equations (DSE) and the functional renormalisation group (fRG). In this setup fRG precision data from arXiv:1706.06326 for the vacuum quark-gluon vertex and gluon propagator of $2$-flavour QCD are used as input, and the respective DSEs are expanded about this input. While the vacuum results for other correlation functions serve as a self-consis…
We put forward a continuum approach for computing finite temperature correlation functions in Yang-Mills theory. This is done in a functional renormalisation group setting which allows for the construction of purely thermal RG-flows. This approach is applied to the ghost and gluon propagators as well as the ghost-gluon vertex in the Landau gauge. We present results in a temperature regime going from vanishing temperature to temperatures far above the confinement-deconfinement temperature Tc. Our findings compare well with the lattice results available.
